We present an extensive analysis on the spatial behavior of the solutions within the three-phase-lags model of a rigid heat conductor for a semi-infinite cylinder excited on its base. The relaxation time of the temperature gradient has a special significance here, namely (i) in the absence of this relaxation time, we manage to highlight a theorem of the domain of influence, that is, outside of a region adjacent to the charged base of the cylinder, all the thermal activity is vanishing, (ii) instead, when the relaxation time of the temperature gradient is present, then it is no longer possible to highlight the area of influence, but we can notice the Saint Venant’s effect. In this latter situation, we are able to describe the exponential decay with respect to the distance from the charged base for a measure of the solution, having a suitable time-dependent exponent to show the rapid decay of the effects when a small time leak has occurred. For the situations when the base of the cylinder is excited for a longer time, both the result expressing the domain of influence and the Saint Venant-type exponential decrease estimate provide insufficient information regarding the spatial behavior along the generator of cylinder. To deal with this shortcoming, we establish exponential decay estimates with a time-independent exponent that can be used to describe the spatial behavior even inside the domain of influence.

我们对在其底座上激励的半无限圆柱体的刚性热导体三相滞后模型中的解的空间行为进行了广泛的分析。温度梯度的弛豫时间在这里具有特殊的意义，即（i）在没有这个弛豫时间的情况下，我们设法强调影响域定理，即在与带电基极相邻的区域之外在圆柱体中，所有热活动都消失了，（ii）相反，当温度梯度的弛豫时间存在时，就不再可能突出影响区域，但我们可以注意到圣维南效应。在后一种情况下，我们能够描述相对于距带电基体的距离的指数衰减，以测量解，并具有合适的时间相关指数来显示当小时间泄漏发生时效果的快速衰减。发生。对于圆柱体底部被激励较长时间的情况，表达影响域的结果和圣维南型指数下降估计都提供了关于沿着圆柱体生成器的空间行为的足够信息。为了解决这个缺点，我们建立了具有与时间无关的指数的指数衰减估计，该指数可用于描述甚至在影响域内的空间行为。